> If you watch slow-motion video of a guitar string vibrating, you’ll see a complex, evolving blend of squiggles. These squiggles are the mathematical sum of all of the string’s different harmonics.

This is incorrect. If you watch a video like [0], the squiggles aren't real, they're an artifact of a rolling shutter camera. A real slowmo camera will correctly show the entire string vibrating[1].

The rest of the article is correct, but you can't see harmonics happening to the string.

[0] https://youtu.be/XOCGb5ZGEV8 [1] https://youtu.be/6sgI7S_G-XI

Actually is not a guitar problem, but all 12-TET tuned instruments have this, it is just a side effect of harmonic math. In the guitar case it is not only the tuning that counts, also the material the string are made and the diameter of the strings count to the final frequency, and we are using parallel frets so applying the same distance to different strings. There are guitars with not parallel frets that try to compensate for the diameter variation. But that’s all math and understanding, cause when you tune your guitar and just play you are in another world were "thought is the killer of flow"; so just play and enjoy the sound. :D

I was born with something not quite like perfect pitch, but when something is even slightly off tune it caused physical discomfort for me.

My cs department had a cool project class where you built what was basically a raspberry pi with a microcontroller by hand, and you had to use the dumb speaker and controller to make your own music firmware to produce notes. the challenge involved, was basically, the processor’s clock wasnt fine grained enough to produce perfect notes. I wanted to make a simon says toy but the notes were off. I approached my professor with my problem and he said I could cheat the processor clock in a clever way to get what i wanted and it was such a “oh wow computers are magic” to me, i got the notes i wanted. disappointingly the TA grader wasnt that impressed but that proff ended up offering me a job before I graduated.

My first guitar teacher told me that someday I'd start to notice that you can't get all strings perfectly in tune. At that point, he said, you'll know you're getting somewhere on the guitar.

This is why string instrument players sometimes prefer to play a note not on the empty string (let's say play a A on the A-string on a cello), but instead on a lower string (e.g. first finger, fourth position on the lower D string) to accord for these imperfections. As a string instrumemt player, you pretty much only have to worry about these notes on empty strings, every other note you can "wiggle into place".

Instruments which have a non-discrete set of pitches (as well as voices) will tend towards the more harmonious (so to speak) tuning when playing in harmony. You’ll notice this in choirs, for example, where singing a capella, the chords will follow nice integral ratios of frequencies. Fretless string instruments and the trombone are obvious cases of instruments which can do micro-tuning, but it’s worth noting that brass instruments have finger loops on some of the valve loops to allow adjustment of pitch. Micro-tuning of the pitch can also be managed in wind instruments through adjustment of the embouchure so while woodwinds seem like they would be only capable of discrete pitches, there is some ability to adjust the pitch during performance.

On a church gig in the 90s, I encountered an organ which was not tuned in equal temperament so that playing guitar with the organ always sounded out of tune (something I only discovered once Mass began since we had rehearsed with a piano) and I had to switch to bass to be able to play an accompaniment that sounded decent.

Not all instruments are limited to a fixed set of pitches. A good classical string player micro-adjusts each individual note to adapt to its harmonic context. For example, making all the thirds and fifths sound good even when the key changes, or adjusting a leading tone up or down very slightly to make it even more leading.

Another way to think of it is that they have to hit every pitch without assistance from the instrument anyway, so they learn to make every note sound “good” rather than hitting a mathematically defined frequency.

A good analogy for equal temperament might be the Gregorian calendar, as a year does not evenly divide perfectly into 365 days. So in order to compensate for that we adjust the calendar by a leap year every so often to make the calendar more accurate in the longer term. That's kinda similar to how every note is a little off in equal temperament so that at the larger scale of being able to play all intervals works out.

> If thirds and fifths are so out of tune in 12-TET, why do we use it? The advantage is that all the thirds and fifths in all the keys are out of tune by the same amount. None of them sound perfect, but none of them sound terrible, either.

Can't we have a system that is optimized for the notes that are actually played in a song rather than the hypothetical set? And what if the optimization is done per small group of notes rather than over an entire song?

The most surprising thing is that everyone here is an expert on true temperament. Who would have guessed.

Article reads like a well akchuallly to is your guitar in tune.

I probably haven't tuned my guitar to concert tuning for a long time.

I tried rocksmith and often tuned to that otherwise I just keep it in tune with itself and what approximately sounds right to me.

My fingers are too fat for any precision to matter too much. So long as it's in tune with itself intonation is vaguely right and the action is acceptable no one will notice my solo playing in the garage by myself is out of tune are the fifth harmonic.

For some reason it’s taken me decades of playing guitar to become good enough at tuning and also sensitive enough to really feel the fact that I can’t tune the guitar. Recently I finally grokked the simple reason that 12 TET cannot be perfect, and it doesn’t take a long article to see it. I was kind-of aware of the major third problem, but I naively thought fifths were still perfect.

A 12 TET chromatic is 2^(1/12), and a 12 TET fifth would be 2^(7/12). A perfect fifth is a 3:2 ratio. Those numbers are slightly different, and that’s enough to understand it. Another way of thinking about it is that if you were to complete the cycle of fifths purely by stacking fifths, you should end up on the note you started with but many octaves higher. But you should be able to see that starting on C1 and going by octaves will produce a number that is purely powers of 2, whereas stacking fifths will necessarily involve powers of both 2 and 3, so they can never be equal, I can stack fifths and never land on my original note’s octaves.

I like the music of Aloboi who makes electronic music, often with just intonation

https://m.youtube.com/watch?v=WkYJb21hpyA

I’m surprised they don’t show true temperament frets.

https://strandbergguitars.com/en-WW/magazine/true-temperamen...

They solve exactly for this issue, and sound amazing in use. The downside is that you are somewhat locked into a given tuning.

Alternatively you can take the approach of guitars with movable frets so you can adjust them per tuning.

https://youtu.be/EZC69A8TsJ8?si=7hUIb7FEKb45eV_L

These are generally used for microtonal playing but can also effectively be true temperament as well.

Guitars with gut frets used to have adjustable positions, which allowed for some mitigation via changing fret positions too

The other problem I always notice on top of all this is that when you pluck a string, it adds tension to it temporarily, so the pitch when you first play it is a little higher than the pitch as it settles down. The louder you play it, the more the effect.

> Not everyone in history thought that 12-TET was an acceptable compromise. Johann Sebastian Bach thought we should use other tuning systems

This is presented as fact, but as I understand it there is no conclusive evidence for what Bach intended wrt temperament. There is a theory that the title page of the Well-Tempered Clavier encodes Bach’s preference in the calligraphic squiggles, but this is a recent theory and speculative. I don’t believe there are any direct statements by Bach as to his intention.

There is no way to tune your guitar so that all the successive open fourths (and the one major third) are pure, without the high E being quite off pitch relative to the low one.

But, unless you mainly play stacked fourths, why would you make it a requirement? You can, for instance, tune instead to get pure fretted fifths between adjacent strings, and fretted octaves between strings one removed.

The real reason you can't get your guitar in tune is one which makes none of the above matter. Most guitars don't have good intonation. Most acoustic guitars don't have movable saddles to set intonation at the bridge. Electric ones do. For accurate tuning, you need not only compensation at the bridge, but also at the nut.

https://guitarnutcompensation.com/

On my main axe, I installed a small screw next to the nut, right under the G string. Just doing the G string makes a huge difference!

Here is a test: play an open D power chord (open D, A on G string, D on B string) it is very clean. Now release the A to play a 1-4-8 G power chord (open D, open G, D).

On my compensated guitar, both of them are crisply in tune. Without nut intonation, one of the two will have ugly beats. If you tune one, the other goes wonky.

When I first heard how good it is after putting in the compensating screw, I was astonished and at the same time filled with the regret of not having done it decades earlier.

Why the G? The unwound G string on electrics is the most susceptible to bad intonation at the nut, because it undergoes the greatest pitch change when it is fretted. Guitarists like to bend that one for the same reason. Fretting it at the first or second frets makes it go markedly sharp; for that reason we need to shorten the distance between the nut and the first fret to get that sharpened interval back down to a semitone.

This is less of a problem on guitars with a wound G, which has a lot more tension in it to compensate for its weight, and doesn't pitch-bend nearly as easily.

I always thought it's an optimization problem: the headstock is pulled by the strings towards the guitar's body, and whenever you get one string in tune, the change in (the distribution of) force changes the tuning of all other strings as well. So ideally they should be moved into an acceptable configuration collectively. People with six hands should be able to do it.

Relevant MinutePhysics video which dives deep: https://www.youtube.com/watch?v=tCsl6ZcY9ag

I thought this was going to be about stiffness of the strings and how the modes even on a single string are not in tune compared to the mathematical model, which assumes infinitely flexible strings.

Even if you tuned two string to ensure that two specific notes on them vibrated at a perfect interval, there are non-multiplicative overtones modulated by resonance with the rest of the instrument. Those intervals are ideals for minimizing dissonance. Practically, the dissonance of 12TET intervals falls below the noise floor of all the other acoustic distortions that give instruments character.

fun fact: some bands, like red hot chilli peppers, will tune the G string slightly flat such that major thirds become just, for some of their riffs. Listen to "scar tissue" for example

Close enough for rock'n roll, as it is called.

I wondered why this article was so high on the front page but now I realize it’s simply because everyone else wanted to “um actually” it. I guess that makes sense.

I read the title and felt personally attacked :(

Interesting.

Advanced banjo players will sometimes use harmonics for a ‘bell’ effect. Here’s a short video from Alison Brown, a great player.

https://www.youtube.com/shorts/NJDgpw2jIdc

Great article. Also, awesome comments; thanks everyone.

Piano tunings are also "stretched" so that the harmonics are more in tune. This is especially needed on verticals and short "baby" grands. See https://en.wikipedia.org/wiki/Stretched_tuning

I have software I use when I tune my Bosendorfer 290 that calculates the stretch. Of course, the final tweaks are done by ear.

Isn’t there a similar issue with pianos?

Well, there's only 6 knobs and if you want to be "in tune with the world" those six knobs can only be in one place.

However if you want more notes than that to be their best you're going to have to compromise and work at it a bit.

Now if you want the instrument to sound its absolute best on its own solo, a slightly different place for some strings.

And then depending on other musicians you are playing with and the way their tuning has achieved perfection (or not), some further tweaking can make a big difference.

And that's after accepting that the "knobs can only be in one place".

For students to get really good at the tuning process can require a few extra years of everyday practice more than it does to learn to play a few pieces.

Part of the limitation is the way only a few minutes of tuning are spent for every hour of practice, if that.

Woah, so cool when a topic I was going into in depth gets to HN.

I'm a relatively new adult beginner on the violin, and one of the fascinating (and extremely difficult) things about un-fretted string instruments is the player has the freedom to shift the tuning around to fit the context. On the violin, we normally play melodies and scales using Pythagorean tuning (which is actually a misnomer as Pythagoras didn't invent it, the ancient Mesopotamians did), which is based on the circle of fifths and leads to wider whole steps and narrower half steps than equal temperment tuning. But then for double stops (i.e. chords), and especially when playing in a string quartet, just intonation, which is based on the harmonic series, is used so the notes sound concordant. This page describes all the different tuning systems a violinist may use, also including 12 TET when trying to match a piano: https://www.violinmasterclass.com/posts/152.

This video shows how challenging it can be when trying to adjust intonation when playing in a string quartet: https://youtu.be/Q7yMAAGeAS4 . Interestingly, the very beginning of that video talks about what TFA discussed that when you tune all your strings as perfect fifths your major thirds will be out of tune.

I'll also put in a plug for light note, an online music theory training tool that was mentioned on HN a decade ago: https://news.ycombinator.com/item?id=12792063 . I'm not related to the owner in any way, I just bought access a few years ago and think it was the first time I really understood Western music theory. The problem with music theory is that the notation is pretty fucked up because it includes all this historical baggage, and lots of music theory courses start with what we've got today and work backwards, while I think it's a lot easier to start with first principles about frequency ratios and go from there.

Other notes (pun intended!): The violin is great for learning music theory because you can actually see on the string how much you're subdividing it - go one third of the way, that's a perfect fifth, go halfway, that's an octave, etc. Harmonics (where you lightly touch a string) are also used all the time in violin repertoire. Finally, the article mentions Harry Patch, but you should also check out Ben Johnston, a composer who worked with Patch and was famous for using just intonation. Here is is Amazing Grace string quartet, and you can really hear the difference using just intonation: https://youtu.be/VJ8Bg9m5l50

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Absurd. A guitar within tolerance is in tune. It's a fundamental feature of the instrument. Not a flaw.

Music doesn't live in an abstract realm of perfections, it is an expression however formed. The fact that we can measure it is one thing. But the music or instruments do not need conform to discrete measurements to satisfy.

I know engineers hate this, but ask any musician. It's like arguing that a sitar and its scales aren't right. Absurd.

Fixed it: “Why can’t you tune your poorly made guitar?”

The most guitars today are still made in the style of the 1950s Gibsons and Fenders, including the neck and tuner layout. Most guitar buyers focus on the aesthetic and not the quality. I switched to a headless guitar where the tuners are at the bridge and it has a fanned fretboard giving the strings more natural tensions, the thing stays in tune and is intonated at the frets extremely well.

Tuning guitar strings by hand is trivial. Try to tune a piano. There have up to 3 strings per key, and quite a lot of keys. Cannot be perfect, but should sound "warm" enough.